Trajectory tracking of under-actuated ships based on optimal sliding mode control with state observer

An optimal sliding mode control (SMC) with state observer is presented for trajectory tracking of under-actuated ships with the model uncertainties, environmental disturbances, input constraints and optimization. Firstly two backstepping virtual control laws are designed by the kinematic model to convert trajectory tracking into heading control and surge velocity control. Then, according to the Norrbin model and the correlation coefficient between surge velocity and propeller revolution, a heading model and a surge velocity model are established respectively. Meanwhile, two extended state observers are built based on the hyperbolic tangent function with clear bounded meaning, which estimate model uncertainties and disturbances. The nonlinear SMC is used to control heading and surge velocity, which embeds the control laws, namely rudder angle and propeller revolution, in a cost function. This cost function is solved with input limits to address the input optimization, amplitude and increment constraints. Finally, the stability analysis is provided, and the simulations show that the ship can track trajectory successfully in case of the disturbances and constraints. These results demonstrate the feasibility and effectiveness of this method.